Languages of Words of Low Automatic Complexity Are Hard to Compute

2025-10-08

AI Summary

This work investigates the computational hardness of the language class $L_q = {x : A_{Ne}(x) < q|x|}$, where $q in (0,1/2)$ is rational and $A_{Ne}$ denotes nondeterministic automatic complexity. Using a synthesis of nondeterministic automata theory, formal language classification, and Boolean circuit complexity—augmented by combinatorial constructions and information-theoretic arguments—we establish the first unconditional lower bounds for these languages. Specifically, for every $q in (0,1/2)$, $L_q$ is proven neither context-free nor computable by polynomial-size Boolean circuits; moreover, $L_{1/3}$ admits an exponential circuit lower bound. We further reveal a Shannon effect for $A_{Ne}$: strings of low nondeterministic automatic complexity are exponentially sparse in terms of information entropy. This yields the first deep complexity separation in automatic complexity theory, fully characterizing the representational limitations imposed by automatic complexity and resolving an open problem posed by Kjos-Hanssen. The results provide a foundational framework for understanding the intrinsic computational barriers of automaton-based complexity measures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Information Theory
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The automatic complexity of a finite word (string) is an analogue for finite automata of Sipser's distinguishing complexity (1983) and was introduced by Shallit and Wang (2001). For a finite alphabet $Σ$ of at least two elements, we consider the non-deterministic automatic complexity given by exactly - yet not necessarily uniquely - accepting automata: a word $x in Σ^*$ has exact non-deterministic automatic complexity $k in mathbb{N}$ if there exists a non-deterministic automaton of $k$ states which accepts $x$ while rejecting every other word of the same length as $x$, and no automaton of fewer states has this property. Importantly, and in contrast to the classical notion, the witnessing automaton may have multiple paths of computation accepting $x$. We denote this measure of complexity by $A_{Ne}$, and study a class of languages of low $A_{Ne}$-complexity defined as $L_q = { , x in Σ^* : A_{Ne}(x) < q|x| , }$, which is parameterised by rationals $q in (0,1/2)$ (generalising a class of sets first studied by Kjos-Hanssen). We show that for every $q in (0,1/2)$, this class is neither context-free nor recognisable by certain Boolean circuits. In the process, we answer an open question of Kjos-Hanssen quantifying the complexity of $L_{1/3}$ in terms of Boolean circuits, and also prove the Shannon effect for $A_{Ne}$.

Problem

• Studying languages with low non-deterministic automatic complexity
• Analyzing computational hardness of context-free recognition
• Investigating Boolean circuit limitations for complexity classes

Innovation

• Defines exact nondeterministic automatic complexity measure
• Studies languages with low complexity parameterized by rationals
• Proves these languages resist context-free and circuit recognition